Rendiconti di Matematica, Serie VIIVolume 27, Roma (2007), 1-16

The “form” of a triangle

ALAIN CHENCINER

Abstract: Heron’s formula, maybe due to Archimedes, expresses the squared areaof a triangle as a polynomial in the squared lengths of its sides. A true understandingof this formula comes from an adequate coding of the shape of a triangle, i.e. of thetriangle up to translations, rotations and reflexions. One can then build a “space oftriangle shapes” which, if orientation is added becomes the “shape sphere” and possessesvery nice symmetries which were recently used in researches on the 3-body problem. Theinvariant interpretation of the shape sphere in turn sheds light on the nature of Heron’sformula.

This is the english version (with an appendix added) of a text in French whichwill appear in a book dedicated to the memory of Gilles Chatelet. The french title“la forme de n corps”, has a pun, hard to render into English because it playson the two meanings of the french word “forme”: shape and quadratic form.

– Introduction: Heron’s formula

The form is quadratic and the bodies are points, even if of celestial origin.At the beginning, there is the formula of Heron of Alexandria [H] for thearea |Δ| of a triangle whose sides have lengths α, β, γ:

(H) 16|Δ|2 = (α + β + γ)(−α + β + γ)(α − β + γ)(α + β − γ) .

Key Words and Phrases: Heron’s formula – Volumes – Inertias – Dispositions –Invariants.A.M.S. Classification: 51K05 – 34C14 – 15A63 – 70E15

2 ALAIN CHENCINER [2]

The last three factors are easily understood as they vanish when the triangle isflat (the second when the length of the longest side is α, the third if it is β, thefourth if it is γ). If one sets

a = α2, b = β2, c = γ2 ,

the formula becomes

16|Δ|2 = 2ab + 2bc + 2ca − a2 − b2 − c2 .

Surfaces and determinants have much in common and so it is not too astonishingto discover that this formula takes the form of the Cayley determinant

(C) −16|Δ|2 = det

⎛⎜⎝0 1 1 11 0 c b1 c 0 a1 b a 0

⎞⎟⎠ .

Nevertheless, a 2× 2 determinant would be less surprising(1) than a 4× 4 deter-minant. What follows originates from work ([AC], see also [A3] and [C1]) donein collaboration with Alain Albouy on the symmetries of the n-body Problemand their reduction. That work generalizes Lagrange’s fundamental memoir [L]to more than three bodies. In it one understands that the above determinantis simply a means of computing a subtler determinant, namely that of an en-domorphism of a two-dimensional vector space which possesses no privilegedbasis. (Think of the plane of equation x + y + z = 0 in IR3. This plane is nat-urally equipped with the three lines of intersection with the coordinate planesx = 0, y = 0 and z = 0 but certainly not with a canonical basis.) Choices ortricks are necessary to compute in such a plane.

1 – The “dispositions” and the reduction of translations

Let us consider a triangle in the plane IR2 or the space IR3. Its area is notaffected by a rigid motion or by a symmetry. This is the fundamental propertythat we want to exploit. It explains why the area depends only on the lengthsof the sides and not on the absolute positions of the vertices.

(1)Indeed, if X is the matrix whose columns are the components of the two vectors�V , �W ∈ IR2, det X is the oriented area of the parallelogram generated by the two vectorsand dettXX is the squared area.

[3] The “form” of n bodies 3

Fig. 1: Invariance of the area by translation.

The way to compute “up to translations” goes back to the works of Jacobion the n-body Problem [J]. What follows is a formalization of his ideas. Givingn points �r1, �r2, . . . , �rn in the space E ≡ IRk, k = 1, 2, 3, . . . is the same as givingthe linear mapping X : IRn → E defined by X(ξ1, ξ2, . . . , ξn) =

∑ni=1 ξi�ri. In

other words, one represents the “configuration” of n points (bodies) in E by thek×n matrix, whose ith column consists of the coordinates of �ri in E ≡ IRk. Thisrepresentation was introduced by Alain Albouy in his PhD thesis [A1].

The source IRn of the mapping X represents the “side of the bodies”, thetarget E represents the “side of space”.

Now, let D∗ be the subspace of IRn which consists of the n-tuples (ξ1, ξ2, . . . ,ξn) whose sum equals 0:

D∗ =

{(ξ1, ξ2, . . . , ξn) ∈ IRn,

n∑i=1

ξi = 0

}.

Whatever be �t ∈ E and (ξ1, ξ1, . . . , ξn) ∈ D∗, we have

n∑i=1

ξi(�ri + �t) =n∑

i=1

ξi�ri .

Hence, the restriction x of X to D∗ no longer distinguishes the two n-tuples(�r1, �r2, . . . , �rn) and (�r1 +�t, �r2 +�t, . . . , �rn +�t). Yet, it gives the differences �ri −�rj

and hence the positions of the bodies once the position of one of them has beenfixed. It follows that giving n points up to a translation in E is the same asgiving the mapping x : D∗ → E.

Such a mapping can be represented by the k × n matrix whose columnsconsist in the components of the vectors �ri ∈ IRk but as well by the matrixwhose columns consist in the components of the vectors �ri + �t, where �t ∈ IRk.It is only after we choose a basis of D∗ that we can get a representation by a(k − 1) × n matrix.

4 ALAIN CHENCINER [4]

But where does D∗ come from? One has to interpret the subspaceD∗ as the dual of the space D of n-tuples of points on the real line up to atranslation. This latter space, called the disposition space, is the quotient of IRn

by the line generated by the vector (1, 1, . . . , 1): whatever be t ∈ IR, the n-tuples(x1, x2, . . . , xn) and (x1 + t, x2 + t, . . . , xn + t) represent the same element inD. To (ξ1, ξ2, . . . , ξn) ∈ D∗ and (x1, x2, . . . , xn) representing an element of D,the duality associates the well defined real number

∑ξixi =

∑ξi(xi + t). A

homomorphism x : D∗ → E may then be interpreted as an element of the tensorproduct D⊗E. The representation of x by an equivalence class of k×n matricescorresponds to the one of D⊗E as the quotient of IRn⊗E ≡ En by the diagonalaction of the translations in E.

2 – From the “side of space” to the “side of the bodies”: the reductionof rotations and symmetries

The invariance under translations of the area |Δ| means that it dependsonly on x; its invariance under linear isometries (i.e. rotations and symmetrieswith respect to a vector subspace) means that it depends in fact only on theGram matrix, whose coefficients are the scalar products 〈�ri, �rj〉E . In particular,if x is represented by the k × n matrix X whose columns are the components ofthe vectors �ri ∈ E ≡ IRk, |Δ| depends only on the n × n matrix tXX.

The Gram matrix must of course be interpreted as the matrix of a quadraticform β on D∗ (and not on IRn). This quadratic form is a complete coding forthe shape (“forme” in french) defined by the n points up to isometries; it maybe written:

β(ξ, η) =∑i,j

〈�ri, �rj〉E ξiηj =∑i,j

(−1

2r2ij

)ξiηj ,

where rij = ||�ri − �rj || is the distance between i and j. Notice that the lastequality does not hold on IRn but only on D∗.

Fig. 2: Invariance of the area by rotation and symmetry.

[5] The “form” of n bodies 5

Remark. One can show that the mutual distances rij are the coordinatesof β in a natural basis of the vector space of quadratic forms on D∗. Hencegiving these is the same as giving β and no mention has to made of the ambientspace E. We have passed from “the side of space” to “the side of the bodies”.

From the definition of β one deduces that β(ξ, ξ) = ||∑i ξi�ri||2E ≥ 0. Thefollowing theorem states that the converse holds (see [Bo], [AC]):

Theorem (Borchart 1866). The real numbers rij are the mutual distancesof n points in an euclidean space E (whose dimension is not imposed) if and onlyif the quadratic form β =

∑i,j

(− 1

2r2ij

)ξiηj on D∗ is non negative. Moreover,

on can choose E of dimension k if and only if the rank of β is less than or equalto k.

This theorem was rediscovered several times. See for example [S] and [Bl].

3 – Masses and volumes

Let us transform the points �ri into “bodies”, possibly “celestial bodies”, byassigning positive masses mi to them. A classical way of reducing the translationsymmetry is to fix at the origin of IRn the center of mass xG = (1/

∑mi)

∑mixi

of an n-tuple (x1, x2, . . . , xn). Doing so, one identifies D to the hyperplane ofIRn with equation

∑mixi = 0. In restricting to this hyperplane the mass scalar

product (or kinetic energy scalar product) defined on IRn by the formula

(x1, x2, . . . , xn) · (y1, y2, . . . , yn) =n∑

i=1

mixiyi ,

one turns D into an Euclidean space whose scalar product may be written

(x1, x2, . . . , xn) · (y1, y2, . . . , yn) =n∑

i=1

mi(xi − xG)(yi − yG) .

To this euclidean structure we associate the isomorphism

μ : D → D∗, μ(x1, x2, . . . , xn) =(m1(x1−xG), m2(x2−xG), . . . , mn(xn−xG)

)which endows D∗ with the euclidean scalar product

(ξ1, ξ2, . . . , ξn) · (η1, η2, . . . , ηn) =n∑

i=1

1mi

ξiηi .

6 ALAIN CHENCINER [6]

Using this scalar product we can turn the quadratic form β into a symmetricendomorphism B of the euclidean space D∗, defined by

β(ξ, η) = B(ξ) · η = ξ · B(η) .

We come back to the case of 3 bodies, where dimD∗ = 2. We can give nowan interpretation of the 4 × 4 determinant in formula (C) as an artefact in thecomputation of the determinant of an endomorphism of a two-dimensional spacewithout privileged basis.

Proposition. The formula (C), which expresses the squared area |Δ|2 ofthe triangle with vertices �r1, �r2, �r3 as a Cayley determinant, is equivalent to theequation

det B =m1m2m3

m1 + m2 + m3(2|Δ|)2 .

Proof. The proof is a sequence of exercises in elementary linear algebra, thetricks of which were announced in the introduction. After a possible translation,one can assume that the center of mass of the �ri lies at the origin of IR2.

1) One defines the extension B of B to an endomorphism of IR3 by the con-dition that B sends to 0 the vector (m1, m2, m3) (this vector generates theorthogonal of D∗ for the scalar product dual to the mass scalar product on(IR3)∗ ≡ IR3. Show that the matrix of B in the canonical basis of IR3 is

B = (bij)1≤i,j≤3, bij = mi < �ri, �rj > .

2) Working in the basis of IR4 = IR× IR3 formed by (1, 0, 0, 0), (0, m1, m2, m3),(0, a1, b1, c1), (0, a2, b2, c2), where the (ai, bi, ci), i = 1, 2, generate D∗, showthat

det B = − 1M

det B ,

where M =∑n

i=1 mi is the sum of the masses and B is the 4 × 4 matrixobtained by adding on top of B the line ( 0 1 1 1 ) and on the left thecolumn ( 0 m1 m2 m3 ).

3) Use the identities

< �ri, �rj > − < �r1, �rj > − < �ri1, �r1 >=< �ri1, �rj1 > ,

< �ri, �rj > −12‖�ri‖2 − 1

2‖�rj‖2 = −1

2r2ij ,

to conclude, with the help of line and column operations, that on the onehand

det B = det

⎛⎜⎝0 1 1 1

m1 0 0 0m2 0 m2 < �r21, �r21 > m2 < �r21, �r31 >m3 0 m3 < �r31, �r21 > m3 < �r31, �r31 >

⎞⎟⎠

[7] The “form” of n bodies 7

is the product by −m1m2m3 of the squared area of the parallelogram gen-erated by the vectors �r21, �r31, and that on the other hand

det B = det

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 1 1

m1 0 −12m1r

212 −1

2m1r

213

m2 −12m2r

221 0 −1

2m2r

223

m3 −12m3r

231 −1

2m3r

232 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠equals the product by 1

4m1m2m3 of the determinant which appears in for-mula (C).

4) Deduce the proposition by recalling that the area of any of the parallelo-grams generated by the �ri equals twice the area of the triangle they define.

5) More generally, show that

det(IdD∗ − λB) = 1 − Iλ +m1m2m3

m1 + m2 + m34|Δ|2λ2 ,

where I = traceB = 1M (m2m3r

223 + m3m1r

231 + m1m2r

212) = 1

M (m2m3a +m3m1b + m1m2c) is the moment of inertia of the three masses with respectto their center of mass.

Remark. All this generalises with the same proof to the case of n bodiesin a finite dimensional euclidean space (exercise). The characteristic polynomialof B is given by the formula

det(IdD∗ − λB) = 1 − η1λ + · · · + (−1)n−1ηn−1λn−1 ,

ηk−1 =1M

∑i1<···<ik

mi1 · · ·mik[(k − 1)!voli1···ik

]2, M =n∑

i=1

mi ,

where voli1···ikis the (k−1)-dimensional volume of the simplex with vertices the

bodies �ri1 , . . . , �rik. In particular, the squared (n − 1)-dimensional volume V of

the simplex whose vertices are the n points is given by the formula

det B =m1 · · ·mn

M

((n − 1)!V

)2, or

(−1)n2n−1(n − 1)!2V 2 = det

⎛⎜⎜⎜⎜⎜⎝0 1 . . . 11 0 . . . .. . . . r2

ij .. . . . . .. . r2

ji . 0 .1 . . . . 0

⎞⎟⎟⎟⎟⎟⎠ .

8 ALAIN CHENCINER [8]

4 – Back to ambient space: inertias

The map x which represents the n-body configuration (�r1, �r2, . . . , �rn) up to atranslation is the restriction to the hyperplane D∗ of the linear map X : IRn → Edefined by X(ξ1, ξ2, . . . , ξn) =

∑ni=1 ξi�ri. The transposed map sends the dual

E∗ of E into the dual of D∗, which is canonically identified with D. But theeuclidean structures of E = IRk and D give an identification of each of thesespaces with its dual. Finallly, the transposed tx : E → D∗ of x may be definedby the formula:

∀e ∈ E,∀ξ ∈ D∗, e · x(ξ) =t x(e) · ξ ,

where the scalar products · are respectively taken in E and in D∗.The endomorphism B : D∗ → D∗ now becomes B =t x ◦ x. But the

endomorphism I = x ◦t x : E → E is well-known by mechanicians, at leastin dimension three where the bivectors can be identified with vectors once anorientation has been chosen. It is the dual inertia form of the rigid body definedby the n point masses (which one supposes to be held rigidly to each other bymassless rods). Indeed, if n = k = 3, �ri = (xi, yi, zi) and

∑i mi�ri = 0, x and tx

may be represented respectively by the matrices

X =

⎛⎝ x1 x2 x3

y1 y2 y3

z1 z2 z3

⎞⎠ and tX =

⎛⎝ m1x1 m1y1 m1z1

m2x2 m2y2 m2z2

m3x3 m3y3 m3z3

⎞⎠ ,

and the extension B of B to IR3 introduced in Paragraph 3 becomes B = tXXwhile

I = XtX =

⎛⎜⎜⎜⎜⎜⎝

∑k

mkx2k

∑k mkxkyk

∑k mkxkzk∑

k

mkykxk

∑k mky2

k

∑k mkykzk∑

k

mkzkxk

∑k mkzkyk

∑k mkz2

k

⎞⎟⎟⎟⎟⎟⎠ .

Let us call J : ∧2E∗ → ∧2E the inertia operator, which turns the instantaneousrotation of a rigid body motion of the configuration x into its angular momentum.After identifying its source and target with IR3, it is represented by the matrix:

J =

⎛⎜⎜⎜⎜⎜⎝

∑k

mk(y2k + z2

k) −∑k mkxkyk −∑

k mkxkzk

−∑

k

mkykxk

∑k mk(z2

k + x2k) −∑

k mkykzk

−∑

k

mkzkxk −∑k mkzkyk

∑k mk(x2

k + y2k)

⎞⎟⎟⎟⎟⎟⎠ = (traceI)Id−I .

In particular, the knowledge of the spectrum of any one of the three operatorsB, I,J implies that of the other two. Fixing B is therefore equivalent to fixing up

[9] The “form” of n bodies 9

to rotation the inertia ellipsoid of the configuration. Hence B deserves the nameof intrinsic inertia. Defined on the side of the bodies and no more on the side ofambient space, it is invariant under the group O(IRk) of linear isometries of E =IRk and covariant under the group O(D) of linear isometries of D (the so-called“democracy group” of physicists); on the contrary, the inertia of mechanicians(I or J , which define the inertia ellipsoid in IRk) is invariant under O(D) andcovariant under O(IRk).

Remark. Once the masses are given, the newtonian potential functionU(�r1, �r2, . . . , �rn) =

∑i<j

mimj

rijdepends only on the rij , i.e. on β. Let us call

the isospectral manifold the set of all quadratic forms β on D∗ such that thecorresponding endomorphism B has a given spectrum; this manifold consists ofthe set of shapes whose inertia ellipsoid is fixed up to an isometry of E. Onedefines in [AC] the balanced configurations (= configurations equilibrees) of npositive masses to be the critical points of the restriction of U to an isospectralmanifold. One shows that these are exactly the configurations which admit ahomographic motion in some space E of arbitrary dimension (recall that a homo-graphic motion is one along which the shape does not change up to similarity).This is another example of how ambient space is forgotten. These configurationsgeneralise the classical central configurations which share the same property butonly in a space E of dimension less or equal to 3. Indeed, the central configu-rations are the critical points of the restriction of U to the set of configurationswhose moment of inertia I = trace B with respect to the center of mass is fixed.

5 – The 3-body problem in the plane: from area to oriented area andfrom Heron’s formula to the theory of invariants

In the quadrant IR3+ consisting of triplets (a, b, c) of non negative real num-

bers, the triplets which represent the squared side lengths of a triangle are thosewhich belong to the cone of equation 2ab + 2bc + 2ca − a2 − b2 − c2 ≥ 0. Thisfollows from Borchart’s theorem above, as this inequality is equivalent to thenon negativity of the quadratic form β defined on D∗ by

β(ξ, η) = −12(aξ2η3 + bξ3η1 + cξ1η2) .

Fixing arbitrarily positive masses, this amounts to the non negativity of the traceand the determinant of the corresponding endomorphism B. As a, b, c ≥ 0, thisis equivalent to the stated condition. One can say that the positivity of the righthand side of Heron’s formula embodies the three triangle inequalities.

We now fix the size of the triangles by imposing that their moment of inertiawith respect to the center of mass be equal to 1:

I = trace B = m2m3a + m3m1b + m1m2c = 1 .

10 ALAIN CHENCINER [10]

The intersection of this affine plane with the cone above is an elliptic domain T(a disc if the three masses are the same): it parametrizes the set of triangles withfixed inertia up to an isometry. The boundary of this domain corresponds to theflat triangles, whose area is zero. It contains three marked points, the collisionpoints which label the degenerate triangles with two coincident vertices (fig. 3).

When the ambient space E is of dimension 2, one can take the quotientby the rotations (i.e. the linear isometries which preserve orientation). Thisleads to the space of oriented triangles with fixed inertia: it is a sphere S (theshape sphere), obtained by gluing along their boundaries two copies of T whichcorrespond to the two possible orientations. We indicate in what follows a moreconceptual way of getting this sphere and the structures which are naturallyassociated to it.

Fig. 3: In IR3.

The first remark is that we can now enrich the notion of area by attachingto it a sign which depends on the orientation of the triangle. We shall denote byΔ this oriented area. Analytically, Δ = 1

2�u1 ∧ �u2, where �u1 is the vector withorigin at the first body and extremity at the second and �u2 is any vector withorigin on the segment between the two first bodies and extremity at the thirdbody (if masses are attributed to the bodies and if the origin of �u2 is the centerof mass of the the first two bodies, �u1 and �u2 are the Jacobi coordinates in thespace Hom(D∗, IR2) ≡ D⊗ IR2 ≡ D2 of planar three-body configurations modulotranslation; these coordinates can be obtained by choosing an orthogonal basisin D∗).

The equation (C), which expresses Δ2 as a function of the squared mutualdistances a = r2

23, b = r231, c = r2

12, defines a quadratic cone in IR4 (coordinatesa, b, c,Δ) or a half-cone C if one is interested only in the quadrant a ≥ 0, b ≥0, c ≥ 0. The shape sphere S identifies with the set of generatrices of C, i.e. withthe quotient of C \ {0} by homotheties (fig. 4).

It follows that the cone C and the sphere S appear respectively as a realiza-tion of the quotient of D2 by the action of rotations or by the action of oriented

[11] The “form” of n bodies 11

similarities (rotations and homotheties). To make this point more precise, let usnotice that the action of the group SO(2) of rotations of IR2 endows the spaceHom(D∗, IR2) ≡ D ⊗ IR2 ≡ D2 of planar 3-body configurations modulo transla-tion with the structure of a vector space on the field C of complex numbers, themultiplication by i corresponding to the rotation by +π

2 in IR2 ≡ C. In otherwords, if (u, v) ∈ D2, i(u, v) = −(v, u).

Fig. 4: In IR4.

To go further on, one has to get rid of the choice of coordinates and, forthis, give an intrinsic interpretation of the space IR4 that we have introduced andof the cone C that it contains. The key remark is that the squared mutual dis-tances a, b, c and the oriented area Δ are quadratic functions with real values onthe vector space D2, i.e. functions which in any system of linear coordinates onD, are expressed as second degree polynomials in these coordinates. Moreover,these functions are invariant under rotation: they are quadratic invariants underthe action of the rotation group SO(2). Now, it is classical in algebraic geometryto characterize a space by the space of functions that one can define on it. In thecase we are interested in, the problem of understanding the quotient D2/SO(2) ofD2 by the action of the group SO(2) of complex numbers with modulus 1 is equiv-alent to the problem of understanding the set of real polynomials P : D2 → IRinvariant under this action, i.e. of polynomials P such that P (λv) = P (v) for allλ ∈ C with modulus 1. But any such polynomial can be expressed as a polyno-mial in the quadratic invariant polynomials. Indeed, if one chooses a basis of D2

on the field of complex numbers, i.e. if one identifies D2 with C2 (say by the choiceof Jacobi coordinates associated with a choice of masses), the action of SO(2)becomes λ · (z1, z2) = (λz1, λz2) and a polynomial P (z1, z2) =

∑aijklz

i1z

j1z

k2 zl

2

is invariant if and only if aijkl = 0 ⇒ i− j + k − l = 0. One deduces that P is apolynomial in the quadratic invariants |z1|2, |z2|2,Rez1z2, Imz1z2. It follows thatwe need only understand the quadratic invariants, i.e. the real four-dimensionalvector space Q(D2) generated by the above functions. This space is of courseindependant of the choice of a basis in D2. As quadratic invariants are enough

12 ALAIN CHENCINER [12]

to construct all the polynomial invariants, the quotient D2/SO(2) injects intothe space Q(D2), more precisely into its dual Q(D2)∗, by the evaluation mape : D2 → Q(D2)∗ which to v ∈ D2 associates the linear form q �→ q(v).

But the image of this map is easy to determine. Choose a complex ba-sis of D2 whose coordinates are noted z1, z2. Then Q(D2) is generated by|z1|2, |z2|2,Rez1z2, Imz1z2, and the image C of D2 is defined by the sole quadraticrelation |Rez1z2|2 + |Imz1z2|2 = |z1|2|z2|2, to which must be added the inequal-ities |z1|2 ≥ 0, |z2|2 ≥ 0. It is more pleasant to replace these last coordinatesby

w0 = |z1|2 + |z2|2, w1 = |z1|2 − |z2|2, w2 = 2Re(z1z2), w3 = 2Im(z1z2) ,

because the half-cone C, defined by the equations −w20+w2

1+w22+w2

3 = 0, w0 ≥ 0,appears as the light cone in Minkowski space. In order to identify it with thehalf-cone defined by (C), it remains to notice that a, b, c,Δ form another basisof Q(D2): Heron’s formula expresses simply the quadratic relation satisfied bythe elements of this basis.

Remarks.

1) We have just seen that the image of the map H : C2 → IR4 defined by

(z1, z2) �→ (|z1|2 + |z2|2, |z1|2 − |z2|2, 2Re(z1z2), 2Im(z1z2))

is the half-cone defined by the equations −w20 + w2

1 + w22 + w2

3, w0 ≥ 0. Itscomposition H = π0 ◦H : C2 → IR3 with the projection π0 parallel to w0 onthe subspace IR3 generated by w1, w2, w3 is the classical Hopf map, whichsends the unit sphere of C2 onto the unit sphere of IR×C by the even moreclassical Hopf fibration.

2) The space of generatrices of the half-cone above is the intrinsic definition(independent in particular of any choice of the masses) of the shape sphere.One shows (see [M]) that it inherits a well defined conformal structure bynoticing that, independently of any choice of coordinates, one can definethe notion of circle as a set of generatrices of the half-cone contained in athree-dimensional vector subspace (if one knows the circles, one knows inparticular the infinitesimal circles and one deduces a notion of angle, i.e. aconformal structure). One obtains riemannian metrics in this conformalclass (i.e. defining the same notion of angle) by considering the shape sphereas the set of oriented isometry classes classes of triangles whose moment ofinertia with respect to their center of mass is equal to 1. To check this, itis enough, once the masses are given, to choose a basis on C of D2 comingfrom an orthogonal basis of D∗ (Jacobi type coordinates). For example, forthe metric corresponding to equal masses, one can take z1 = 1√

2(�r2 − �r1)

and z2 =√

23 (�r3 − 1

2 (�r1 + �r2)). In any case, if one has chosen a basis which

[13] The “form” of n bodies 13

is orthonormal, the metric on D2 ≡ C2 ≡ IR4 is the standard euclideanmetric.

3) Topologically, the oriented situation is much richer than the non orientedone: the shape sphere possesses a symmetry group with twelve elements(the dihedral group D6, which is also the symmetry group of the regularhexagon). Deprived of the three collision points, it acquires a fundamentalgroup isomorphic to the free group on two generators. Recent works haveshown that part of this richness shows up in the periodic solutions of the 3body problem, but this is another story (see [C2]).

– Appendix. Darboux’s interpretation of the quadratic form β

In [D], Gaston Darboux gives the following interpretation of β in the casewhen it is non degenerate i.e. when it describes a non degenerate n-simplex inIRn−1 (he considers only the case n = 4 but this is immaterial; compare toProposition 14 of [A2]). To each point �r in IRn−1 he attaches its barycentrichomogeneous coordinates (ξ1, · · · , ξn) with respect to the given simplex, defined(as an element of the projective space) by(

n∑i=1

ξi

)�r =

n∑i=1

ξi�ri ,

where �r1, · · · , �rn are the vertices of the simplex. This amounts to identifyingIRn−1 with the hyperplane T = 1 in IRn(coordinates X, Y, . . . , T ) and call-ing ξ1, . . . , ξn the coordinates of any point on the line generated by (�r, 1) =(x, y, . . . , 1) in the basis {(�r1, 1), . . . , (�rn−1, 1)} of IRn. Hence

�r =(

X

T,Y

T, . . .

)∈ IRn−1, X =

n∑i=1

ξixi, Y =n∑

i=1

ξiyi, . . . , T =n∑

i=1

ξi .

He then notices that in such coordinates the sphere circumscribed to the simplex(defined by X2+Y 2+· · ·−R2T 2 = 0 if we suppose that the center �s of this sphereis at the origin of IRn−1 and call R its radius) has equation

∑i,j r2

ijξiξj = 0. Adirect proof is easily found; one can also, as explained to me by Martin Celli,use Huyghens formula for the momentum of inertia:

∀�s ∈ IRn−1,

n∑i=1

ξi|�s − �ri|2 =n∑

i=1

ξi|�r − �ri|2 + (n∑

i=1

ξi)|�s − �r|2 .

Choosing as �s the center of the circumscribed sphere, the formula becomes

−n∑

i=1

ξi|�r − �ri|2 =

(n∑

i=1

ξi

)(X2

T 2+

Y 2

T 2+ · · · − R2

),

14 ALAIN CHENCINER [14]

which implies that an equation of this sphere is∑n

i=1 ξi|�r − �ri|2 = 0 (recallthat for the points not at infinity,

∑ni=1 ξi = 0). But, by the very definition of

barycentric coordinates,∑n

i=1 ξi(�r − �ri) = 0 and it follows from a formula ofLeibniz that

−12

n∑i,j=1

ξiξjr2ij = −

⎛⎝ n∑j=1

ξj

⎞⎠ n∑i=1

ξi|�r−�ri|2 =

(n∑

i=1

ξi

)2(X2

T 2+

Y 2

T 2+ · · · − R2

),

that is

−12

n∑i,j=1

ξiξjr2ij = X2 + Y 2 + · · · − R2T 2 ,

so that the equation of the circumscribed sphere is∑n

i,j=1 ξiξjr2ij = 0.

In particular, the quadratic form β(ξ, ξ) appears as the restriction of theequation of this sphere to the hyperplane at infinity.

Let us see now how Darboux deduces the formula for the squared volume ofan n-simplex (see end of Section 3) from the consideration of the “contravariant”of the triple formed by the equation of the circumscribed sphere X2 +Y 2 + · · ·−R2T 2 = 0 (in coordinates with the origin at the center of the sphere) and twicethe linear form T = 0 defining the “hyperplane at infinity”. This means that hededuces the formula from the computation of the determinants of both sides ofthe following identity (the notations are those of Section 3 and the identity isobvious either directly or as a consequence of what is proved in 3):

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 1 . . 11 0 . . . .

1 . 0 . −12r2ij .

. . . . . .

. . −12r2ji . 0 .

1 . . . . 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=

=

⎛⎜⎜⎜⎝1 0 0 . 00 x1 y1 . 10 x2 y2 . 1. . . . .0 xn yn . 1

⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝

0 0 . . 10 1 . . 00 0 . . .0 . . 1 01 0 . . −R2

⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝

1 0 . . 00 x1 x2 . xn

0 y1 y2 . yn

0 . . . .0 1 1 . 1

⎞⎟⎟⎟⎠ .

This identity expresses the transformation of the “contravariant” formed by thequadratic form and the two linear forms under the action of the linear groupGL(n, IR) through its natural extension to GL(n+1, IR) in which it acts only on

[15] The “form” of n bodies 15

the last n coordinates. In the same way, one deduces immediately the formula

det

⎛⎜⎜⎜⎝0 . . . .. . . r2

ij .. . . . .. r2

ji . 0 .. . . . 0

⎞⎟⎟⎟⎠ = (−1)n+12n((n − 1)!V R

)2

from the identity ⎛⎜⎜⎜⎜⎜⎜⎝

0 . . . .

. . . −12r2ij .

. . . . .

. −12r2ji . 0 .

. . . . 0

⎞⎟⎟⎟⎟⎟⎟⎠ =

=

⎛⎜⎝x1 y1 . 1x2 y2 . 1. . . .

xn yn . 1

⎞⎟⎠⎛⎜⎝

1 0 . . 00 1 . . .0 . . 1 00 0 . . −R2

⎞⎟⎠⎛⎜⎝

x1 x2 . xn

y1 y2 . yn

. . . .1 1 . 1

⎞⎟⎠ .

Acknowledgements

Thanks to Alain Albouy for references [Bo], [D], to Daniel Meyer for refer-ence [S] and to Richard Montgomery for correcting my English. Thanks to ItaloCapuzzo Dolcetta and Piero Negrini for inviting me to speak on this topic in themath colloquium of La Sapienza. Thanks to Charles Alunni, the editor of thevolume dedicated to Gilles Chatelet, for the permission to publish this englishversion.

REFERENCES

[A1] A. Albouy: Integral manifolds of the N-body problem, Inventiones Mathematicæ,114 (1993), 463-488.

[A2] A. Albouy: On a paper of Moeckel on Central Configurations, Regular andChaotic Dynamics, 8 (2003), 133-142.

[A3] A. Albouy: Mutual distances in Celestial Mechanics, Lectures at Nankai Insti-tute, Tianjin, China, 2004.

[AC] A. Albouy – A. Chenciner: le probleme des n corps et les distances mutuelles,Inventiones Mathematicæ, 131 (1998), 151-184.

[Bl] L. M. Blumenthal: Theory and applications of distance geometry , Oxford atthe Clarendon press, 1953.

16 ALAIN CHENCINER [16]

[Bo] C. W. Borchart: Ueber die Aufgabe des Maximum, welche der Bestimmung desTetraeders von grosstem Volumen bei gegebenem Flacheninhalt der Seitenflachenfur mehr als drei Dimensionen entspricht , Memoires de l’academie de Berlin,pp. 121-155, Werke pp. 201-232 (19 fevrier 1866); the theorem appears on page207.

[C1] A. Chenciner: Introduction to the N-body problem, Ravello summer school, 1997,available at http://www.imcce.fr/Equipes/ASD/person/chenciner/chen preprint.html.

[C2] A. Chenciner: De l’espace des triangles au probleme des 3 corps, Gazette desmathematiciens, 104 (2004), 22-38.

[D] G. Darboux: Sur les relations entre les groupes de points, de cercles et de spheresdans le plan et dans l’espace, Annales scientifiques de l’E.N.S., I serie 2eme (1872),323-392.

[H] Heron of Alexandria: Metrica, (-124). See Heath, Greek mathematics p. 420-422. The formula is written as S2 = s(s−α)(s−β)(s−γ), where s is the half-sumof the side lengths : 2s = α + β + γ.

[J] C. G. J. Jacobi: Sur l’elimination du nœud dans le probleme des trois corps,C.R. Acad. Sci. de Paris, XV (1842), 236-255, in Gesammelte werke, band IV,pp. 295-316. The formulae which appear at the beginning of this paper amount tothe choice of an orthogonal basis in D.

[L] J. L. Lagrange: Essai sur le probleme des trois corps, Prix de l’Academie Royaledes Sciences de Paris, IX (1772), in œuvres, 6 (1772), 229-324.

[M] R. Montgomery: Infinitely Many Syzygies, Arch. Rational Mech. Anal., 164(2002), 311-340.

[S] I. J. Schoenberg: Remarks to Maurice Frechet’s article “Sur la definition ax-iomatique d’une classe d’espaces distancies vectoriellement applicable sur l’espacede Hilbert”, Annals of Mathematics, 36 (1935), 724-732.

Lavoro pervenuto alla redazione il 16 maggio 2006ed accettato per la pubblicazione il 13 giugno 2006.

Bozze licenziate il 31 gennaio 2007

INDIRIZZO DELL’AUTORE:

Alain Chenciner – Astronomie et Systemes Dynamiques – IMCCE, UMR 8028 du CNRS – 77,avenue Denfert-Rochereau – 75014 Paris, France – etDepartement de Mathematiques – Universite Paris VII-Denis Diderot – 16, rue Clisson – 75013Paris, France